state space representation Block Diagrams state space representation transfer function representation Assume that we are only interested in the input-output relation: transfer function Then, a system can be represented by a block with an input and output. One can represent a large system as an interconnection of block diagrams of subsystems. ... gives an insight both in analysis and synthesis.

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Reduction Rules of Block Diagrams + + - + + + + -

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- + + + + + Find the transfer function by reducing the block diagram!

A block diagram of a circuit is given A block diagram of a circuit is given. Find the transfer function for each block and obtain the transfer function of the block diagram Vd3/Vk!

Signal-Flow Diagram A diagram equivalent to block diagrams. Suitable to apply Mason’s rule – a formula that gives the transfer function. The signal-flow diagram is a directed graph (digraph) set of vertices set of edges with weights on edges, which represent gains (transfer functions) between components inputs and outputs.

Reduction of Signal-Flow Diagrams x1 x2 a x1 x2 a1 x3 a2 x1 x3 a1 a2 x1 x2 a b x1 x2 a+b x3 x4 c x2 a x1 b x4 x2 a x1 ac bc

x1 x2 a x3 b c x1 x3 x1 ab x3 bc x1 a x4 x2 x3 d b c e x1 a x4 x2 x3 e

Block Diagrams vs. Signal-Flow Diagrams G(s) R(s) C(s) R(s) C(s) G(s) R(s) E(s) 1 C(s) G(s) -1 G(s) R(s) C(s) E(s) + - Mason’s Rule Aim: To find the transfer function directly from the diagram without any reduction.

Definitions: Path: A subgraph GP of the graph G is called a path if it contains n edges and n+1 vertices, one can label its edges as e1, e2, ...,en and its vertices as v1,v2, ....,vn+1 such that the edge ek points from vk to vk+1. Path gain: The product of gains along a path. Loop: A subgraph GL of the graph G is called a loop if it contains n edges and n vertices, one can label its edges as e1, e2, ...,en and its vertices as v1,v2, ....,vn such that the edge ek points from vk to vk+1(mod n). Loop gain: The product of gains along a loop. Determinant of a (sub)graph: loop gains products of loop gains for disjoint loop pairs products of loop gains for disjoint loop triples

Mason’s Rule For a signal-flow diagram the transfer function between the input and output vertices is given by Gain of the k. path Determinant of the subgraph obtained by deleting the k. path Determinant of the whole diagram